Nowadays, computer-driven numerical simulation is becoming increasingly popular as an integral part of all areas of research and engineering. Simulations can often be performed faster, cheaper, and in a potentially safer manner than benchtop experiments. Describing the real-world with systems of equations also helps us to verify the existence of different phenomena, predict future behavior, and discover new idiosyncrasies in complex systems.
In our work, we study how biological cells respond to externally-applied electric fields using numerical simulations based on the finite element method (FEM) for the specific purpose of cancer treatment. FEMs are based on the concept of approximating the geometry of a real-world problem using smaller two-dimensional shapes (often triangles or quadrilaterals) or three-dimensional solids (often tetrahedra). A complex global problem may be reduced to a number of local problems on each of these smaller geometries—called elements—and each of these elements is related to those surrounding it, forming a mesh of elements representing the global geometry and a large system of coupled equations. For the specific application reported here, the geometries of biological cells are described in two dimensions using triangles that are refined around high-aspect ratio features, such as at the cell and nuclear membranes. The cells in our simulation need to be described mathematically as accurately as possible, to render appropriate spatial, temporal, and physical characteristics. Typically, many simplifications are made to a cell’s morphology in order to more easily represent it computationally, which include simple shapes in the form of circles/spheres and ellipses/ellipsoids and a limited amount of physical phenomena. However, biological cells are complex structures with ever-changing geometry and physics that span many length and time scales. To truly capture and explore physical phenomena in such a system requires accounting for more of the complexities of a biological cell, such as its irregular geometry.
Our cover image for the November 15 issue of Biophysical Journal was created during a project where we showed the importance of using realistic cell shapes when studying electric field exposure. We used fluorescence microscopy images to extract realistic cell shapes and convert them into a two-dimensional numerical model. The image shows a triangular FEM mesh, fit to the cell boundaries, where each system of equations is generated on each triangular element that describes their relation to other neighboring elements. Each element is also assigned material parameters, such as electrical conductivity and permittivity. The cell and nuclear membrane boundaries are discretized using numerous boundary nodes to resolve the irregular geometry and minimize the numerical error introduced into the calculation, while retaining sufficient solution efficiency. Conversely, fewer nodes are needed in the middle of the nuclear regions for example, where material boundaries are not present. The above mesh consists of almost one million triangles in a rectangular computational domain that measures around 200 microns across. The image is a zoom of a large model system with realistic cell geometries, containing 91 tightly packed cells and nuclei. The nuclei were highlighted in blue during the assignment of parameters to the nuclear regions in the software used. A glowing effect was added during post-processing for a bit of a dramatic flare.
These types of simulations are not just for research purposes though. Historically, we have utilized robust FEM models to predict how electric currents will propagate in a tissue. Such models are also utilized during treatment planning for tumor ablation procedures in which a series of electrical pulses are delivered to the tumor via two or more electrodes to maximally destroy malignant cells while preserving critical stromal components, such as vasculature. More realistic models of biological cells allows treatments to be delivered to the appropriate malignant region and further limit the damage to the surrounding healthy tissue.
More simulations like this are sure to follow that include detailed geometries, extend these and similar models to three dimensions, and account for additional relevant biophysical phenomena. Though simplifications must be made to any simulation, the trend of increasing computer power and performance enables many of these inhibitions to be overcome by simulations that achieve a greater predictive capacity and come ever closer to simulating precise experimental and clinical conditions.
—Tomo Murovec and Daniel C. Sweeney